1. Field of the Invention
The present invention relates to a computer simulation method for the electrical characteristics of semiconductor devices and, more particularly, to a numerical simulation method for the transient response characteristics of junction capacitance.
2. Description of the Related Art
According to a conventional computer simulation method for the electrical characteristics of semiconductor devices, as described in Ryo Dan, ed., xe2x80x9cTechniques for Process Device Simulationsxe2x80x9d, pp. 91-134, the region to be analyzed is divided into a mesh pattern, and the Poisson equation, electron current continuity equation, and hole current continuity equation are discretized at the respective mesh points. Further, these equations are linearized to obtain simultaneous linear equations by, e.g., the Newton method. These simultaneous linear equations are calculated to obtain solutions.
In addition to this general device simulation, there are methods of calculating PN junction capacitance and Schottky junction capacitance in a device by a numerical simulation, such as small-AC-signal analysis described in, e.g., Section xe2x80x9cSinusoidal Steady-State Analysis (S3A) xe2x80x9d, p. 2,032 in Steven E. Laux, xe2x80x9cTechniques for Small-Signal Analysis of Semiconductor Devicesxe2x80x9d, IEEE Trans. Electron Devices, Vol. 32, No. 10, pp. 2,028-2,037. This analysis will be explained briefly. In the small-AC-signal analysis, junction capacitance or the like is calculated to obtain a DC solution at a desired operating point. More specifically, equations (1) to (3) below are equivalent to equations (3.41)-(3.43) in xe2x80x9cTechniques for Process Device Simulationsxe2x80x9d, pp. 105-106 cited earlier:
F("psgr",n,p)=0xe2x80x83xe2x80x83(1)
G("psgr",n,p)=0xe2x80x83xe2x80x83(2)
H("psgr",n,p)=0xe2x80x83xe2x80x83(3)
where
F("psgr",n,p)=∇(xcex5xc2x7∇"psgr")+q(pxe2x88x92n+NDxe2x88x92NA)xe2x80x83xe2x80x83(4)
                              G          ⁡                      (                          ψ              ,              n              ,              p                        )                          =                                            ∂              n                                      ∂              t                                +                                    G              0                        ⁡                          (                              ψ                ,                n                ,                p                            )                                                          (        5        )            xe2x80x83G0("psgr",n,p)=∇(xcexcnn∇"psgr"xe2x88x92Dn∇n)xe2x88x92GR(n,p)xe2x80x83xe2x80x83(6)
                              H          ⁡                      (                          ψ              ,              n              ,              p                        )                          =                                            ∂              p                                      ∂              t                                +                                    H              0                        ⁡                          (                              ψ                ,                n                ,                p                            )                                                          (        7        )            xe2x80x83H0("psgr",n,p)=xe2x88x92∇(xcexcpp∇"psgr"+Dp∇p)xe2x88x92GR(n,p)xe2x80x83xe2x80x83(8)
are solved for                                           ∂            n                                ∂            t                          =        0                            (        9        )                                                      ∂            p                                ∂            t                          =        0                            (        10        )            
to calculate the potential, electron density, and hole density by
"xgr"0=("psgr"0,n0,p0)xe2x80x83xe2x80x83(11)
where "psgr" is the electrostatic potential, n is the electron density, p is the hole density, xcex5 is the permittivity, q is the unit charge, ND is the donor density, NA is the acceptor density, xcexcn and xcexcp are the electron and hole mobilities, Dn and Dp are the electron and hole diffusion constants, and GR is the generation/recombination term of an electron/hole. Assume that a small AC bias with an angular frequency xcfx89 and an amplitude {tilde over (V)} is superposed on a DC bias V at the operating point, and the potential, electron density, and hole density in this state respond in a manner given by
"xgr"="xgr"0+{tilde over ("xgr")} exp jxcfx89txe2x80x83xe2x80x83(12)
{tilde over ("xgr")}=({tilde over ("psgr")},xc3x1,{tilde over (p)})xe2x80x83xe2x80x83(13)
Substituting the response in equation (12) into equations (1) to (8) and Taylor-expanding each term using
∥{tilde over ("xgr")}∥ less than ∥"xgr"0"xgr"
and
F("psgr"0,n0,p0)=0xe2x80x83xe2x80x83(14)
G0("psgr"0,n0,p0)=0xe2x80x83xe2x80x83(15)
H0("psgr"0,n0,p0)=0xe2x80x83xe2x80x83(16)
yields equations (17) below equivalent to equation (10) in xe2x80x9cTechniques for Small-Signal Analysis of Semiconductor Devicesxe2x80x9d, p. 2,032                                          (                                                                                                      ∂                      F                                                              ∂                      ψ                                                                                                                                  ∂                      F                                                              ∂                      n                                                                                                                                  ∂                      F                                                              ∂                      p                                                                                                                                                              ∂                                              G                        0                                                                                    ∂                      ψ                                                                                                                                  j                      ⁢                                              xe2x80x83                                            ⁢                      ω                      ⁢                                              xe2x80x83                                            ⁢                      t                                        +                                                                  ∂                                                  G                          0                                                                                            ∂                        n                                                                                                                                                        ∂                                              G                        0                                                                                    ∂                      p                                                                                                                                                              ∂                                              H                        0                                                                                    ∂                      ψ                                                                                                                                  ∂                                              H                        0                                                                                    ∂                      n                                                                                                                                  j                      ⁢                                              xe2x80x83                                            ⁢                      ω                      ⁢                                              xe2x80x83                                            ⁢                      t                                        +                                                                  ∂                                                  H                          0                                                                                            ∂                        p                                                                                                                  )                    ⁢                      xe2x80x83                    ⁢                      (                                                                                ψ                    ~                                                                                                                    n                    ~                                                                                                                    p                    ~                                                                        )                          =                  (                                                                      V                  ~                                                                                    0                                                                    0                                              )                                    (        17        )            
By solving equation (17), the complex amplitude {tilde over ("xgr")} of potential, electron density, and hole density is obtained. Assuming that each element of the coefficient matrix is an equivalent conductance, the conductance current component of a response AC current at a device electrode is obtained by the product of the equivalent conductance coupled to the device electrode and defined on a mesh branch, and the complex amplitude across the mesh points on the ends of the branch. By adding the displacement current component   ϵ  ⁢            ∂              (                  ∇          φ                )                    ∂      t      
to the conductance current component, the total response AC current Ĩ is obtained. Using this, the capacitance component viewed from the device electrode is given by                     C        =                              I            ~                                ω            ⁢                          xe2x80x83                        ⁢                          V              ~                                                          (        18        )            
FIG. 1 is a flow chart showing the processing procedure of this method. In step 301, a DC bias to be applied to a device is set. In step 302, internal physical quantities such as potential, electron density, and hole density at the current operating point are calculated by steady-state analysis. In step 303, the frequency of a small RF voltage to be applied to the device is set. In step 304, a perturbation equation for a small RF AC component is solved to obtain the complex amplitude of a small AC response component of the internal physical quantity. In step 305, the small AC component of an electrode current is calculated from the complex amplitude obtained in step 304, and the junction capacitance viewed from the electrode is calculated using the result. Instep 306, whether AC analysis is complete for all frequencies set in advance is checked. In step 307, whether analysis is complete for all DC bias voltages set in advance is checked.
The above description concerns simulation for junction capacitance in a steady state. In addition, there is an experimental method of measuring transient changes in junction capacitance, such as DLTS described in, e.g., Takashi Katoda, ed., xe2x80x9cEvaluation Techniques for Semiconductorsxe2x80x9d, pp. 245-247. According to this method, after a step voltage pulse is applied to a device electrode, a small RF voltage of about 1 MHz is applied to the device, and the phase difference with an RF current flowing at that time is continuously observed to measure temporal changes in junction capacitance. No concrete conventional technique has been proposed about this experimental numerical simulation. However, transient changes in capacitance can be numerically simulated by applying the Fourier analysis of transient analysis results which is described in Section xe2x80x9cFourier Decomposition of Transient Excitations (FD)xe2x80x9d, pp. 2,029-2,030 in Steven E. Laux, xe2x80x9cTechniques for Small-Signal Analysis of Semiconductor Devicesxe2x80x9d, IEEE Trans. Electron Devices, Vol. 32, No. 10, pp. 2,028-2,037. More specifically, similar to an actual experiment, an input waveform prepared by superposing a step voltage pulse on a small RF voltage is applied to a device electrode, and transient changes in capacitance are calculated using the transient response current waveform by transient analysis of the device simulation. In transient analysis, an equation describing changes in charged state of a deep impurity level that cause transient changes in junction capacitance is added to equations (1) to (7). These equations are discretized including the time differential terms. The response waveform is divided by time windows using the reciprocal of the frequency of the small RF voltage. Fourier analysis is performed in each window to obtain the average capacitance component in each window. The average capacitance components are combined to obtain final temporal changes in capacitance.
FIG. 2 is a schematic view showing the outline of this method. FIG. 2 schematically shows the case wherein a voltage waveform 402 prepared by superposing a small RF AC voltage on a pulse voltage is directly input and subjected to transient analysis, and a junction capacitance 405 of a device 401 is obtained by Fourier analysis 404 of a response current waveform 403 and the input waveform 402.
The above method of numerically simulating the transient response of capacitance by transient analysis and subsequent Fourier analysis has a problem of a large calculation amount. In general, the transient response time constant of capacitance that is processed by DLTS measurement and the like falls within the range of several msec to several sec. To the contrary, capacitance measurement is generally performed by an RF voltage of 1 MHz. To perform Fourier analysis with satisfactory precision using a time window of about 10xe2x88x926 sec, transient analysis must be performed at a time step width of about 10xe2x88x927 sec. Therefore, simulating all transient responses requires about 104 to 107 analysis points, resulting in a long calculation time.
The present invention has been made to solve the conventional drawbacks, and has as its object to provide a computer simulation method for semiconductor devices in which the transient response of junction capacitance in a semiconductor device can be numerically simulated with a small calculation amount at a high speed.
To achieve the above object, according to the main aspect of the present invention, there is provided a computer simulation method for a semiconductor device, comprising the steps of (a) obtaining, by transient analysis, a temporal change in internal physical quantity including an electrostatic potential, an electron density, and a hole density in a semiconductor device upon application of a pulse voltage, (b) performing AC signal analysis by inputting a small RF AC voltage, assuming various physical quantities obtained at each time are in a pseudo steady state, and (c) calculating junction capacitance in the semiconductor device, the steps (a), (b), and (c) being repeatedly performed until a predetermined analysis time is reached to obtain a transient temporal change in junction capacitance.
According to the method of the present invention, after changes in physical quantities in a device upon application of a pulse voltage are obtained by transient analysis, small-AC-signal analysis is performed assuming the physical quantities are in a pseudo steady state at each analysis time. In the conventional method, a small RF voltage waveform is actually input to perform transient analysis. Accordingly, 104 to 107 analysis points are necessary to obtain a transient response with a time constant of 10xe2x88x923 to 1 sec. In the present invention, however, since the capacitance value can be obtained by one AC analysis for each time, the total number of analysis points can be suppressed to such a degree as to express the transient response of capacitance, e.g., to 10 points per transient response time constant, which is two to three orders of magnitude smaller than the conventional case. Assumption of a pseudo steady state in the present invention gives a good approximation as far as the response time constant of junction capacitance is much larger than the reciprocal of the frequency of the small RF voltage, and no problem occurs under normal experimental conditions.